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Generative Rate-Distortion-Perception Theory on Riemannian Manifolds

Tong Wu, Zhiyong Chen, Meixia Tao, Dazhi He, Feng Yang

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Abstract

Generative communication uses transmitted information to control stochastic generation at the receiver. A fundamental question is how much source-specific information must be communicated so that outputs generated at the receiver satisfy prescribed fidelity and reliability requirements while preserving the conditional source distribution. We address this question for sources on compact Riemannian manifolds by developing a generative rate-distortion-perception (GRDP) framework without common randomness. We formulate an asymptotic operational GRDP function under a geodesic distortion radius and a failure-probability constraint, and establish its single-letter characterization through an equivalent informational formulation. Furthermore, for the informational formulation with the high-reliability limit, we derive a geometric lower bound determined by the largest source probability mass of a region with bounded geodesic diameter, together with an upper bound based on probabilistic coverings of the source manifold. Small-radius analysis of the lower bound reveals the roles of intrinsic dimension and scalar curvature in the required communication rate. For uniformly distributed sources on compact homogeneous Riemannian manifolds, we prove that the bounds coincide, yielding an exact characterization of the limiting GRDP function. We further specialize this result to unit spheres, obtaining an explicit expression that relates the minimum communication rate to the geodesic distortion radius and intrinsic dimension, revealing that for the dd-dimensional unit sphere with d≥2d\ge2, increasing the communication rate by dd bits per source symbol can reduce the distortion radius by more than half.

Introduction

Recent advances in generative artificial intelligence have made it increasingly feasible to model natural data distributions and generate high-quality samples from learned models [1, 2, 3, 4, 5]. These developments have motivated generative communication, in which transmitted information controls a stochastic generation process at the receiver [6, 7, 8, 9, 10]. This paradigm shifts the emphasis from reconstructing a source representation through conventional decoding to conveying source-specific conditions that guide generation at the receiver. In this paradigm, the learned source prior captures statistical regularities shared across source realizations, while the transmitted message conveys information about the particular realization to be communicated. The receiver conditions the generative model on this message and samples an output using local randomness [11], as illustrated in Figure 1. This approach has been explored for image transmission [12, 13], video delivery [14, 15, 16], immersive media [17, 18], and extended-reality services [19].

Generative communication model on a Riemannian manifold

Figure 1. Generative communication model on a Riemannian manifold under conditional-distribution preservation. The black point on the right serves as the source reference for illustrating the geodesic distortion dg(xt,x^t)d_g(x_t,\hat{x}_t).

These developments call for an information-theoretic framework that characterizes the fundamental limits of generative communication. A key question is how much source-specific information must be communicated for outputs generated at the receiver to satisfy prescribed fidelity and reliability requirements while preserving the conditional source distribution. Answering this question requires characterizing the minimum communication rate under joint fidelity, reliability, and conditional-distribution constraints.

The rate-distortion-perception (RDP) framework provides a natural starting point [20], as it jointly characterizes communication rate, reconstruction fidelity, and distributional realism. It extends classical rate-distortion theory by introducing a perception constraint that measures the discrepancy between the source and reconstruction distributions. Subsequent work established coding theorems for the RDP function and clarified the role of randomness in encoding and decoding [21, 22]. In particular, under symbol-wise perception constraints, the operational RDP function admits a single-letter characterization without common randomness [22]. Under sequence-level marginal perception constraints, however, obtaining a general characterization without common randomness is more challenging. Existing studies have therefore exploited additional structure on the perception constraint, distortion measure, or source distribution.

One line of work considers the perfect-perception setting, in which the reconstruction is required to preserve the source distribution [22, 23, 24]. These studies establish single-letter characterizations under the corresponding operational criteria and examine the role of randomness. Another line exploits specific distortion and perception measures to obtain tractable characterizations. For example, [25] establishes a single-letter characterization under squared error distortion and a squared Wasserstein-2 perception measure with limited common randomness, together with an explicit evaluation for Gaussian sources. In addition, [26] develops copula-based methods for estimating the perfect-perception RDP function of continuous sources and derives analytical lower bounds under mean squared error (MSE) distortion.

While these studies primarily constrain marginal distributions, [27] investigates a conditional-distribution-based perception measure that compares the source and reconstruction distributions conditioned on the encoder output. Building on the posterior reference map introduced in [28], it establishes a single-letter characterization under sequence-level conditional perception constraints without common randomness for discrete memoryless sources. For continuous memoryless sources, it further establishes a characterization in Euclidean spaces under squared-error distortion and a squared Wasserstein-2 perception measure, including the perfect-perception case. For Gaussian vector sources, the resulting characterization admits a reverse water-filling-type solution. These results provide a foundation for analyzing how transmitted information constrains the distribution of generated outputs.

Despite these advances, existing RDP characterizations do not directly address the joint role of intrinsic source geometry and generation reliability under conditional-distribution preservation. Extending these results to the setting considered here requires addressing two challenges.

First, many tractable RDP characterizations for continuous sources are developed in Euclidean spaces with squared-error distortion. Natural media sources, however, may exhibit low-dimensional geometric structure within a high-dimensional ambient space. The manifold hypothesis suggests that their distributions can concentrate on or near low-dimensional manifolds [29, 30, 31, 32]. For example, although facial images are represented by high-dimensional pixel vectors, much of their variation can be described by a small number of factors, such as pose, expression, and illumination [29]. This motivates modeling structured sources on Riemannian manifolds and measuring fidelity using intrinsic geodesic distance. Although such sources can also be represented in an ambient Euclidean space, ambient distance need not capture their intrinsic geometry. A geometric characterization is therefore needed to make explicit how the intrinsic dimension and curvature of the source manifold affect the required communication rate.

Second, conditional-distribution consistency and generation reliability impose distinct requirements. In generative communication, the receiver samples an output from a conditional distribution specified by the received information. Even when this distribution matches the conditional source distribution exactly, an individual generated output need not be close to the actual source realization. Different outputs sampled under the same received information can therefore have substantially different distortions. Average-distortion criteria do not explicitly quantify the probability of exceeding a prescribed distortion threshold. The relevant question is thus how much communication is required to satisfy a prescribed geodesic distortion tolerance with a target failure probability while preserving the conditional source distribution.

Motivated by these considerations, we extend the Euclidean source model to sources taking values on Riemannian manifolds and measure fidelity using the geodesic distance induced by the Riemannian metric. To capture the reliability of stochastic generation, we adopt a geodesic excess-distortion criterion [33], which quantifies the probability that the distortion between the source and the generated output exceeds a prescribed threshold. Together with conditional-distribution preservation, this criterion allows us to characterize the tradeoff among communication rate, geodesic distortion radius, and generation failure probability, thereby quantifying the fidelity and reliability achievable through communication-controlled generation.

The contributions of this paper can be summarized as follows:

  • We formulate an asymptotic operational generative rate-distortion-perception (GRDP) function on Riemannian manifolds under exact conditional-distribution preservation and without common randomness. The formulation incorporates a prescribed geodesic distortion radius and an excess-distortion probability to capture the fidelity and reliability of stochastic generation. In Theorem 1, we establish an equivalent single-letter informational

characterization, thereby identifying the minimum asymptotic communication rate required to control generation under these constraints.

  • We derive geometric lower and upper bounds on the informational GRDP function in the high-reliability regime in Theorem 2 and Theorem 3, respectively. The lower bound relates the necessary communication rate to the maximum source probability contained in a region of bounded geodesic diameter. The upper bound provides an achievable construction based on probabilistic coverings by such regions, with overlapping regions offering flexibility in representing the source. A small-radius analysis further makes explicit how the intrinsic dimension and scalar curvature contribute to the rate lower bound.

  • We establish an exact characterization of the informational GRDP function in the high-reliability limit for uniformly distributed sources on compact homogeneous Riemannian manifolds in Theorem 4. Specifically, we show that the geometric lower bound is achievable by randomizing over isometric images of admissible regions. The minimum communication rate is thus determined by the largest source probability mass of a region satisfying the prescribed geodesic diameter constraint. We further specialize this result to unit spheres, obtaining an explicit expression that relates the required rate to the geodesic distortion radius and intrinsic dimension, revealing that for the dd-dimensional unit sphere with d≥2d \ge2, increasing the communication rate by dd bits per source symbol can reduce the distortion radius by more than half.

The remainder of this paper is organized as follows. Section II introduces the source and system models, motivates the distortion and perception criteria, and defines the operational and informational GRDP functions under conditional-distribution preservation. Section III establishes the equivalence of these functions, yielding a single-letter characterization, and derives lower and upper bounds that reveal the role of intrinsic source geometry. Section IV establishes the tightness of these bounds for homogeneous manifolds and illustrates their evaluation for spherical sources. Section V concludes the paper.

Problem Statement

Geometric Source Model

As shown in Figure 1, we consider to characterize the fundamental limits of generative communications, which transmits the continuous-valued natural data from the real world. As suggested in [29, 30, 31, 32], the real-world data may concentrate on low-dimensional manifolds rather than occupying the entire ambient Euclidean space. Therefore in this case, it is necessary to explicitly account for the intrinsic geometry of the source space and investigate its impact on communication rate. To this end, we model the support of the source distribution as a Riemannian manifold M\mathcal{M} with intrinsic dimension dd, embedded in Rm\mathbb{R}^{m}, with d<md < m.

The manifold M\mathcal{M} is then equipped with the necessary geometric structures.

  • Tangent Space TxMT_{\mathbf{x}}\mathcal{M}: Denotes the tangent space of M\mathcal{M} at a point x\mathbf{x}, which is a dd-dimensional linear vector space.

  • Riemannian Metric gg: Let gg be a Riemannian metric on M\mathcal{M}. At each point x∈M\mathbf{x} \in\mathcal{M}, gxg_{\mathbf{x}} is a positive-definite symmetric bilinear form on the tangent space TxMT_{\mathbf{x}}\mathcal{M}.

  • Scalar Curvature S(x)S(\mathbf{x}): Denotes the scalar curvature of the manifold at point x\mathbf{x} induced by gg, which quantifies the infinitesimal bending of the local space at that point.

  • Volume measure dVgdV_g: Induced by the Riemannian metric gg with respect to the standard Euclidean coordinate volume element dx=dx1⋯dxdd\mathbf{x}=dx^{1}\cdots dx^{d} is given by:

dVg=det⁡g dx,(1)dV_g = \sqrt{\det g}\,d\mathbf{x}, \tag*{(1)}

  • Volume of a region Vg(O)V_g(O): For any Borel measurable subset O⊆MO \subseteq\mathcal{M}, its Riemannian volume is defined as the Lebesgue integral of the volume measure over the target space OO:

Vg(O)=∫OdVg.(2)V_g(O) = \int_O dV_g. \tag*{(2)}

  • Curves Space Ω(x1,x2)\Omega(\mathbf{x}_1,\mathbf{x}_2): Denotes the space of all piecewise smooth parameterized curves on M\mathcal{M} connecting two points x1\mathbf{x}_1 and x2\mathbf{x}_2 on the manifold.

  • Riemannian length L(γ)L(\gamma): is defined as the integral of the norm of its tangent vector γ˙(t)=dγdt∈Tγ(t)M\dot{\gamma}(t) = \frac{d\gamma}{dt} \in T_{\gamma(t)}\mathcal{M} under the metric gg for any curve γ∈Ω(x1,x2)\gamma\in\Omega(\mathbf{x}_1,\mathbf{x}_2) in this space,

L(γ)=∫01gγ(t)(γ˙(t),γ˙(t)) dt.(3)L(\gamma) = \int_0^1 \sqrt{g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))}\,dt. \tag*{(3)}

  • Geodesic Distance dg(x1,x2)d_g(\mathbf{x}_1,\mathbf{x}_2): is defined as the infimum of the Riemannian lengths of all admissible curves in Ω(x1,x2)\Omega(\mathbf{x}_1,\mathbf{x}_2) connecting two points x1\mathbf{x}_1 and x2\mathbf{x}_2 on the Riemannian manifold:

dg(x1,x2)=inf⁡γ∈Ω(x1,x2)L(γ).(4)d_g(\mathbf{x}_1,\mathbf{x}_2) = \inf_{\gamma\in\Omega(\mathbf{x}_1,\mathbf{x}_2)} L(\gamma). \tag*{(4)}

  • Geodesic Ball Br(x)B_r(\mathbf{x}): Is formally defined as the geodesic ball centered at a given point x\mathbf{x} with a radius r>0r > 0:

Br(x)={x^∈M∣dg(x,x^)≤r}.(5)B_r(\mathbf{x}) = \{\hat{\mathbf{x}} \in\mathcal{M} \mid d_g(\mathbf{x},\hat{\mathbf{x}}) \leq r\}. \tag*{(5)}

Besides the basic structure, the manifold M\mathcal{M} should be connected and compact, meaning that ∀x1,x2∈M\forall\mathbf{x}_1,\mathbf{x}_2 \in\mathcal{M}, the geodesic distance is attained and

dg(x1,x2)<∞,Vg(M)<∞∀x1,x2∈M(6)d_g(\mathbf{x}_1,\mathbf{x}_2) < \infty,\quad V_g(\mathcal{M}) < \infty\quad\forall\mathbf{x}_1,\mathbf{x}_2 \in\mathcal{M} \tag*{(6)}

The compactness assumption is motivated by the bounded range of practical natural data, while the connectedness can be imposed without loss of generality by restricting attention to a connected component of the data space.

With the mathematical definition, we can now define the probability density of natural data X\mathbf{X} by the Radon-Nikodym derivative as

f(x):=dPXdVg(x)(7)f(\mathbf{x}) := \frac{dP_{\mathbf{X}}}{dV_g}(\mathbf{x}) \tag*{(7)}

with its Lebesgue integral ∫Mf(x)dVg(x)=1\int_{\mathcal{M}} f(\mathbf{x})dV_g(\mathbf{x}) = 1, where PXP_{\mathbf{X}} is a probability measure on M\mathcal{M} that is absolutely continuous with respect to the Riemannian volume measure VgV_g.

In conclusion, in this paper, we consider the case that the support of source X\mathbf{X} is a connected and compact Riemannian manifold and its density is therefore naturally defined on the manifold by Radon-Nikodym derivative. Moreover, it is obvious that the quantities related to the probability should be defined on the manifold M\mathcal{M} by Lebesgue integral with respect to the Riemannian volume measure VgV_g. For example, the differential entropy of X\mathbf{X} is defined as

h(X)=−∫Mf(x)log⁡2f(x)dVg.(8)h(\mathbf{X}) = -\int_{\mathcal{M}} f(\mathbf{x})\log_2 f(\mathbf{x})dV_g. \tag*{(8)}

Remark 1. This Riemannian manifold model directly characterizes the geometry of the source support, rather than treating the source merely as a random variable in the ambient Euclidean space. Moreover, it generalizes the conventional Euclidean source model, since the Euclidean space is itself a special case of a Riemannian manifold.

Generative Communications Model and Problem Definition

Let {Xt}t=1∞\{\mathbf{X}_t\}_{t=1}^{\infty} denote a stationary and memoryless source with marginal distribution PXP_{\mathbf{X}} on a connected and compact manifold M\mathcal{M}. A stochastic encoder maps a length-nn source sequence Xn\mathbf{X}^n to a message EE in a binary prefix code E\mathcal{E} through the conditional distribution PE∣XnP_{E\mid\mathbf{X}^n}. At the receiver, a generator samples a length-nn sequence X^n\hat{\mathbf{X}}^n according to the conditional distribution PX^n∣EP_{\hat{\mathbf{X}}^n\mid E}.

We next analyze the characteristics of generative communication to identify the requirements that the theoretical framework should address. First, the generator is modeled to approximate the conditional source distribution given the transmitted information. The theoretical framework should therefore incorporate a conditional-distribution constraint. Under suitable regularity conditions, modern conditional generative models, such as conditional diffusion models, provide theoretical guarantees that the learned conditional distribution can approximate the target conditional distribution arbitrarily well as the amount of training data and the model capacity increase, including in settings involving Riemannian manifolds [34, 35]. This approximation capability is independent of the communication constraints. Accordingly, we consider the idealized case in which the generator perfectly learns the conditional source distribution, meaning that

PX^n∣E=PXn∣Ealmost surely (a.s.).(9)P_{\hat{\mathbf{X}}^n\mid E} = P_{\mathbf{X}^n\mid E}\quad\text{almost surely (a.s.)}. \tag*{(9)}

Second, conditional-distribution preservation also implies that both the source sample and the generated sample lie on the same low-dimensional manifold. We therefore measure fidelity using the geodesic distance dg(x,x^)d_g(\mathbf{x},\hat{\mathbf{x}}) induced by the Riemannian metric. Moreover, generative communication differs from conventional deterministic reconstruction in that the received message specifies a distribution rather than a unique reconstruction. The same message may lead to different generated samples, and the resulting distortion depends on both the source realization and the private randomness at the receiver. To characterize the reliability of stochastic generation controlled by the transmitted information, we define the geodesic excess-distortion probability as

P(r):=Pr⁡{dg(X,X^)≥r},(10)P(r) := \Pr\{d_g(\mathbf{X},\hat{\mathbf{X}}) \geq r\}, \tag*{(10)}

which quantifies the probability that the geodesic distortion is at least a prescribed radius rr.

These observations motivate the following operational formulation of the fundamental limits of generative communication.

Definition 1. For a radius r>0r > 0 and a failure probability 0≤ϵ<10 \le\epsilon< 1, a rate RR is said to be asymptotically achievable under conditional-distribution preservation if there exists a sequence of stochastic block codes indexed by nn, each consisting of a stochastic encoder PE∣XnP_{E|X^n} and a stochastic decoder PX^n∣EP_{\hat{X}^n|E}, satisfying

lim sup⁡n→∞1nE[ℓ(E)]≤R.(11)\limsup_{n\to\infty} \frac{1}{n}\mathbb{E}[\ell(E)] \le R. \tag*{(11)}
lim sup⁡n→∞1n∑t=1nPr⁡{dg(Xt,X^t)≥r}≤ϵ.(12)\limsup_{n\to\infty} \frac{1}{n}\sum_{t=1}^{n}\Pr\{d_g(X_t,\hat{X}_t) \ge r\} \le\epsilon. \tag*{(12)}
PX^n∣E=PXn∣EPE-a.s.(13)P_{\hat{X}^n|E}=P_{X^n|E}\quad P_E\text{-a.s.} \tag*{(13)}

where ℓ(E)\ell(E) denotes the length of EE. The asymptotic operational GRDP function is defined as

RC(r,ϵ):=inf⁡{R≥0:R is achievable for (r,ϵ)}.(14)R_C(r,\epsilon) := \inf\{R \ge0 : R\text{ is achievable for }(r,\epsilon)\}. \tag*{(14)}

To characterize RC(r,ϵ)R_C(r,\epsilon), we introduce the corresponding informational GRDP function.

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